fraction of boys at least as large as assuming is true. Because the number of births is very large, we can use a normal approximation for the fraction of male births , with and , to compute We would have been equally surprised if we had seen 49,581 female births, i.e. , so a frequentist would usually perform a two-sided test, for which the p-value would be . In both cases, the p-value is lower than the significance level, α, of 5%, so the frequentist approach rejects as it disagrees with the observed data. Bayesian approach Assuming no reason to favor one hypothesis over the other, the Bayesian approach would be to assign prior probabilities and a uniform distribution to under , and then to compute the posterior probability of using Bayes' theorem, After observing boys out of births, we can compute the posterior probability of each hypothesis using the probability mass function for a binomial variable, where is the Beta function. From these values, we find the posterior probability of , which strongly favors over . The two approaches—the Bayesian and the frequentist—appear to be in conflict, and this is the "paradox". Reconciling the Bayesian and frequentist approaches Almost sure hypothesis testing Naaman proposed an adaption of the significance level to the sample size in order to control false positives: , such that with . At least in the numerical example, taking , results in a significance level of 0.00318, so the frequentist would not reject the null hypothesis, which is in agreement with the Bayesian approach. Uninformative priors If we use an uninformative prior and test a hypothesis more similar to that in the frequentist approach, the paradox disappears. For example, if we calculate the posterior distribution , using a uniform prior distribution on (i.e. ), we find If we use this to check the probability that a newborn is more likely to be a boy than a girl, i.e. , we find In other words, it is very likely that the proportion of male births is above 0.5. Neither analysis gives an estimate of the effect size, directly, but both could be used to determine, for instance, if the fraction of boy births is likely to be above some particular threshold. The lack of an actual paradox The apparent disagreement between the two approaches is caused by a combination of factors. First, the frequentist approach above tests